Solve the linear programming problem. Maximize and minimize \[ P=6 x+12 y \] subject to \[ \begin{aligned} x+y & \leq 2 \\ x+3 y & \leq 4 \\ 4 x+5 y & \geq 20 \\ x, y & \geq 0 \end{aligned} \] What is the maximum value of $P$ ? Select the correct choice below and fill in any answer boxes present in your choice. A. $P=\square$ (Type an integer or a fraction.) B. There is no maximum value of $P$.

Step 1 ：First, graph the constraints to find the feasible region. The constraints are: \(x+y \leq 2\), \(x+3y \leq 4\), \(4x+5y \geq 20\), and \(x, y \geq 0\).

Step 2 ：Next, find the vertices of the feasible region. These are the points where the constraint lines intersect. The vertices are (0,2), (0,4/3), (4/7, 6/7), and (20/9, 0).

Step 3 ：Substitute these vertices into the objective function \(P=6x+12y\) to find the maximum and minimum values.

Step 4 ：For (0,2), \(P=6(0)+12(2)=24\).

Step 5 ：For (0,4/3), \(P=6(0)+12(4/3)=16\).

Step 6 ：For (4/7, 6/7), \(P=6(4/7)+12(6/7)=24\).

Step 7 ：For (20/9, 0), \(P=6(20/9)+12(0)=40/3\).

Step 8 ：The maximum value of \(P\) is 24.

Step 9 ：\(\boxed{P=24}\) is the final answer.